It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I can't remember where I heard this) that the harmonic series is more or less the only important series with this property.

What other methods besides analytic continuation and zeta regularization exist for summing divergent series? Do they work on the harmonic series? And are there other well-known series which also don't have obvious regularizations?

7 Answers
7

One common regularization method that wasn't mentioned in the Everything Seminar post is to take the constant term of a meromorphic continuation. While the Riemann zeta function has a simple pole at 1, the constant term of the Laurent series expansion is the Euler-Mascheroni constant gamma = 0.5772156649...

It is reasonable to claim that most divergent series don't have interesting or natural regularizations, but you could also reasonably claim that most divergent series aren't interesting. Any function with extremely rapid growth (e.g., the Busy Beaver function) is unlikely to have a sum that is regularizable in a natural way.

Let $w$ be a state on the quotient C$^*$-algebra $\ell_\infty / c_0$ (bounded sequences quotient out convergent to zero sequences). Then the functional
$$
\mathrm{Tr}_w(A) = w ( \{ \frac{1}{\log (1+n)} \sum_{j=1}^n \lambda(n,A) \}_{n=1}^\infty )
$$
is a trace on the ideal of compact operators (on a separable Hilbert space) such that
$\mu(n,A) = O(n^{-1})$, $n \geq 1$. Here $\lambda$ denotes the sequence of eigenvalues of the compact operator $A$ ordered so that the sequence of absolute values $| \lambda |$ is a decreasing sequence, and $\mu$ denotes the sequence of singular values (eigenvalues of the absolute value of $A$). If $A_{\mathrm{harmonic}} = \mathrm{diag}(n^{-1})$ (any diagonal operator with the harmonic series as the diagonal) then $\mathrm{Tr}_w(A_{\mathrm{harmonic}})=1$. This is a regularisation of the harmonic series.

Traces on compact operators, thinking of compact operators as noncommutative generalisations of convergent to zero sequences, form summing procedures on these "noncommutative $c_0$ sequences". The trace $\mathrm{Tr}_w$ above is called a Dixmier trace, after the French mathematician Jacques Dixmier who described it in 1968. It has been popularised by Alain Connes in his version of Noncommutative Geometry (Academic Press, 1994). Dixmier traces are not the only traces on the ideal of compact operators such that
$\mu(n,A) = O(n^{-1})$, and there exist other traces $\varphi$ such that $\varphi(A_{\mathrm{harmonic}}) = 1$. Dixmier traces generalise the zeta function residue regularisation and the high temperature (or short time) heat kernel regularisation.
Thus the zeta function residue regularisation is not the only regularisation possible.

There exist many traces defined on certain ideals besides just the canonical trace on the trace class operators (trace class operators are the noncommutative version of the summable sequences $\ell_1$). Deep results are known about which ideals admit non-trivial traces, which translates as meaning which rates of divergence (of convergent to zero sequences) admit a non-trivial summing procedure. See the book "Singular Traces", De Gruyter 2012 (admission of vested interest: I am one of the authors). The harmonic series fortunately admits a rich non-trivial range of summing procedures. Contrast with $\ell_p$ sequences for $p > 1$ whose associated ideals have no non-trivial traces, and sequences $O(n^{-p})$, $p > 1$, whose associated ideals also have no non-trivial traces.

I'm not allowed to post a comment, but in reply to Michael Lugo's post and as a followup to Scott Carnahan, the prime harmonic series can be regularized in analogy with $1 + 1/2 + 1/3 + 1/4 + \ldots$ "$=$" $\gamma$, giving the Mertens constant. See the prime zeta function for more information.

In this case it's not "meromorphic continuation" as the singularity is logarithmic. This leads to the followup question: is there a practical difference, and is there a general theory for the logarithmic (or even more general, e.g. multiply nested logarithmic) case? The prime zeta function has some interesting properties, such as having a natural boundary of analyticity at $\Re(s) = 0$.

I have a sneaking suspicion that anything that works on 1/2 + 1/3 + 1/5 + ... will probably also work on the harmonic series, although I certainly don't have any hard reasoning to back this up -- just that it doesn't have nice local properties or nice global properties, much like the harmonic series.

But I sort of hope I'm wrong -- I'd be very interested to see what a regularization for this series looks like!

Hmmm. Perhaps I was too strong there. I tend to assume that a summation method should obey a_1+a_2+...=0+a_1+a_2+..., which zeta regularization does not. But I can't make a strong argument for that assumption.
–
David SpeyerOct 29 '09 at 12:27

By the way the proposition that $1+1+1+... = -1/2$ was first made by Euler.
–
AndrewAug 16 '11 at 20:31